Gostic 2020 - Practical Considerations for Measuring Rt - Overview

Summary

Gostic et al. (2020, PLoS Comput Biol 16(12): e1008409) is a methods Perspective on estimating the effective reproductive number $R_t$ from surveillance data. Using synthetic SEIR data it recommends the Cori et al. (2013) instantaneous-$R_t$ estimator for real-time use and shows that the dominant errors come from the generation-interval specification, delays between infection and observation, right truncation, ascertainment and smoothing windows.

Overview

Why this source. The primary reference for the renewal-equation $R_t$ model, Cori, Ferguson, Fraser & Cauchemez (2013, Am J Epidemiol 178(9):1505-12), is paywalled at the publisher and its mirrors block scripted download, so this open-access (CC0) Perspective, which re-derives and stress-tests the Cori estimator, is used as the substitute. Formulae for the Cori posterior are given from the original paper’s well-known result and marked as such.

Key result. (i) Cori et al. accurately tracks the instantaneous $R_t$, even through abrupt changes; Wallinga-Teunis estimates the case (cohort) $R_t$, which leads $R_t$ by about one mean generation interval (about 8 days in the synthetic SEIR) and is biased low at the end of a truncated series; Bettencourt-Ribeiro is biased because it implicitly assumes an exponential generation interval. (ii) Naive lag fixes (backward convolution, shifting) blur or lag $R_t$; deconvolution or a forward observation model in the estimation is better. (iii) Intervals ignore ascertainment, delay and truncation uncertainty.

Motivating PyMC examples

E38 (renewal-equation $R_t$ “as a triangular solve” on a nowcast, plus the spurious “wave is ending” from truncated data) and E90 (hospital thresholds and nowcasts). See Nowcasting Concepts - Delay Hazard Models and the Renewal Equation.

Source Map

Paper section	Covered in
Abstract, Introduction, synthetic data design	this note; Gostic 2020 - Instantaneous vs Case Reproductive Number and the Three Estimators
Instantaneous vs case $R_t$; Cori, Bettencourt-Ribeiro, Wallinga-Teunis (Figs 1-2, S1-S2)	Gostic 2020 - Instantaneous vs Case Reproductive Number and the Three Estimators
Generation interval, renewal equation, serial interval (Fig 3)	Gostic 2020 - Generation Interval and the Renewal Equation
Delays to observation: shift, convolve, deconvolve (Figs 4-5, S3)	Gostic 2020 - Rt Estimation Pipeline and Right-Truncation Bias
Right truncation	Gostic 2020 - Rt Estimation Pipeline and Right-Truncation Bias
Incomplete observation; smoothing windows (Fig 6); Conclusions	Gostic 2020 - Observation Process, Smoothing Windows and Recommendations

Connections

Pairs with McGough 2020 - Nowcasting by Bayesian Smoothing - Overview (nowcast the incidence, then compute $R_t$) and the state-space notes Linear-Gaussian State-Space Models (a latent infection series with forward delays is a state-space model).

Relevance to marketing measurement

The renewal idea (new events = current “infectiousness” times a time-varying rate) parallels adstock-style carry-over: today’s response is a convolution of past exposure with a lag kernel. The warning that backward convolution blurs peaks and that deconvolution or a forward model is needed applies to inferring when a campaign’s effect began.

See Also

Bayesian Structural Time-Series Model · Proper Scoring Rules (CRPS, Log Score, Pinball Loss)